Mathematics are that branch of knowledge which is the most independent of any, and the least liable to uncertainty, difference of opinion, and sceptical doubts. However, uncertainties, differences, and doubts, have arisen here; but then they have been chiefly about such parts of mathematics as fall under the consideration of the logician. For, it seems impossible that a man who has qualified himself duly, should doubt about the justness of an arithmetical, algebraical, or fluxional operation, or the conclusiveness of a geometrical demonstration.
The words point, line, surface, infinitely great, infinitely little, are all capable of definitions, at least of being explained by other words. But then these words cannot suggest any visible ideas to the imagination, but what are inconsistent with the very words themselves. However, this inconsistency has no effect upon the reasoning. It is evident, that all that can be meant by the three angles of a triangle being equal to two right ones, or the parabolic area to of the circumscribing parallelogram, or deduced from these positions, must always hold in future fact; and this, as observed above, is all the truth that any thing can have. In fluxional conclusions it is demonstratively evident, that the quantity under consideration cannot be greater or less by any thing assignable, than according to the fluxional conclusion; and this seems to me entirely the same thing as proving it to be equal.
I cannot presume to suggest any particular methods by which farther discoveries may be made in mathematical matters, which are so far advanced, that few persons are able to comprehend even what is discovered and unfolded already. However, it may not be amiss to observe, that all the operations of arithmetic, geometry, and algebra, should be applied to each other in every possible way, so as to find out in each something analogous to what is already known and established in the other two. The application of the arithmetical operations of division and extraction of roots to algebraic quantities, and of the method of obtaining the roots of numeral equations by approximation to specious ones, as taught by Sir Isaac Newton, have been the sources of the greatest fluxional discoveries.
It is the purport of this and the foregoing section, to give imperfect rudiments of such an art of logic, as is defined above, i.e. as should make use of words in the way of mathematical symbols, and proceed by mathematical methods of investigation and computation in inquiries of all sorts. Not that the data in the sciences are as yet, in general, ripe for such methods; but
